User:A.M.R./sigma-algebra
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Say, 
. 
and 
. Therefore, 
is a σ-algebra over set X, that is closed under complementation and countable unions of its members & also contains empty set.
[edit] Measure
Now, A measure on X is a function which assigns a real number to subsets of X; this can be thought of as making precise a notion of 'size' or 'volume' for sets.
[edit] Random Variable
Random variable is different than the source of randomness or scenarios. Suppose, in case of dice rolling, 
is Ω or set of scenarios. w is each outcomes. Now, rv is what assigns values to different outcomes of an experiment. Here, rv X would be X(w) = w. But, in case of stock market, the abstract variable w can be the different situations of market which is constituted with different factors & may not be fully expressed mathematically. But, the random variable can assign these scenarios some values which can be stock prices.
